The online birth index does not detail parents post 1905 so you may have no option other than to purchase her death certificate.
Nothing like a bit of local knowledge....try the Narrogin Family History Centre, c/o 24 Fox St. NARROGIN WA 6312. Someone in the community may remember them. I noticed a Thomas
MATTHEWS was appointed secretary of the Narrogin Race Club in 1930. There is an error on the electoral roll with their address...it should be Glyde Street not Clyde Street. Please give them
all the information you have to save them doing duplicate searching.

I do a lot of searching for queries in my local area and become quite annoyed when I spend time [gratis] finding information to be told...we knew that

On the1910 electoral roll in Subiaco, Thomas is listed as a bootmaker - the information, or more likely rumor,is that about the time that Antoniette disappeared from her English family, the local cobbler by name of Matthews also disappeared!
What year did she/they disappear and from where? Have you found Thomas Morgan
MATTHEWS on the previous census in the area? I can only find a Thomas Morgan
MATTHEWS born Wales. Gosh a twenty year age gap if his age at death is correct

Australian Electoral Roll
WA 1910
MATTHEWS Antoinetta 486 Hay Street, Subiaco HD
MATTHEWS Thomas Morgan 486 Hay Street, Subiaco Bootmaker
Western Australian Postal Directories
1909, 1910
MATTHEWS Thos bootmaker 43 Peel Street, Jolimont
1949
MATTHEWS T M Narrogin
Cando